Question
(a)A radiation detector is placed close to a radioactive source. The detector does not surround the source.
Radiation is emitted in all directions and, as a result, the activity of the source and the measured count rate are different.
Suggest two other reasons why the activity and the measured count rate may be different.
1. …………………………………………………………………………………………………………………………….
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2. …………………………………………………………………………………………………………………………….
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(b) The variation with time t of the measured count rate in (a) is shown in Fig. 12.1.
(i) State the feature of Fig. 12.1 that indicates the random nature of radioactive decay.
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(ii) Use Fig. 12.1 to determine the half-life of the radioactive isotope in the source.
half-life = ……………………………………….. hours
(c) The readings in (b) were obtained at room temperature.
A second sample of this isotope is heated to a temperature of 500°C.
The initial count rate at time t = 0 is the same as that in (b).
The variation with time t of the measured count rate from the heated source is determined.
State, with a reason, the difference, if any, in
1. the half-life,
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2. the measured count rate for any specific time.
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Answer/Explanation
a)
• emission from radioactive daughter products
• self-absorption in source
• absorption in air before reaching detector
• detector not sensitive to all radiations
• window of detector may absorb some radiation
• dead-time of counter
• background radiation
Any two points.
(b)(i)
curve is not smooth
or
curve fluctuates/curve is jagged
(b)(ii)
clear evidence of allowance for background
half-life determined at least twice
half-life = 1.5 hours
(1 mark if in range 1.7–2.0; 2 marks if in range 1.4–1.6)
(c)
1. half-life: no change
because decay is spontaneous/independent of environment
Question
The product of the pressure p and the volume V of an ideal gas is given by the expression
\(pV =\frac{ 1}{3}Nm<c^ 2>\)
where m is the mass of one molecule of the gas.
(a) State the meaning of the symbol
(i) N,
………………………………………………………………………………………………………………[1]
(ii) \(<c ^2>\).
………………………………………………………………………………………………………………[1]
(b) The product pV is also given by the expression
\(pV = NkT\).
Deduce an expression, in terms of the Boltzmann constant k and the thermodynamic temperature T, for the mean kinetic energy of a molecule of the ideal gas. [2]
(c) A cylinder contains 1.0 mol of an ideal gas.
(i) The volume of the cylinder is constant.
Calculate the energy required to raise the temperature of the gas by 1.0 kelvin.
energy = ………………………………………. J [2]
(ii) The volume of the cylinder is now allowed to increase so that the gas remains at constant pressure when it is heated.
Explain whether the energy required to raise the temperature of the gas by 1.0 kelvin is now different from your answer in (i).
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Answer/Explanation
Ans:
(a) (i) N: (total) number of molecules
(ii) \(<c^2>\): mean square speed/velocity
(b) \(pV = \frac{1}{3}Nm<c^2> = NkT\)
(mean) kinetic energy \(= \frac{1}{2} m<c^2> \)
algebra clear leading to \(\frac{1}{2} m<c^2> = (3/2)kT \)
(c) (i) either energy required \(= (3/2) × 1.38 × 10^{–23} × 1.0 × 6.0^2 × 10^{23}\)
either energy required = 12.5 J (12J if 2 s.f.)
or energy \(= (3/2) × 8.31 × 1.0 \)
or energy\( = 12.5 J \)